1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Matrix of eigenvectors, relation to rotation matrix

  1. Apr 12, 2013 #1
    So I am given [itex]B=\begin{array}{cc} 3 & 5 \\ 5 & 3 \end{array}[/itex]. I find the eigenvalues and eigenvectors: 8, -2, and (1, 1), (1, -1), respectively. I am then told to form the matrix of normalised eigenvectors, S, and I do, then to find [itex]S^{-1}BS[/itex], which, with [itex]S = \frac{1}{\sqrt{2}}\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}[/itex], I get [itex]\begin{array}{cc} 8 & 0 \\ 0 & -2 \end{array}[/itex]. All great and dandy, that's the correct answer, and the text then asks me for the rotation matrix, which I give as [itex]\begin{array}{cc} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}[/itex]. However, when asked how the rotation matrix and S are related, I am clearly stumped; checking the answers, they have used [itex]S = \frac{1}{\sqrt{2}}\begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array}[/itex], which I think is a valid choice, since the eigenvector could have been (1,-1) or (-1,1). But since I made a different choice, I cannot get the answer they are looking for (That S is the rotation matrix for theta is 45).

    Thus, have I made a mistake in my calculation? In my reasoning? In my assumptions? Or is it just that I was unlucky to have picked the way I did and the question wasn't expecting that (I doubt the latter)?
     
  2. jcsd
  3. Apr 12, 2013 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You're just unlucky. The way you chose the eigenvectors, S is a rotation combined with a reflection across the line y=x. As you saw, you can choose the eigenvectors so that S is just a rotation.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Matrix of eigenvectors, relation to rotation matrix
  1. Rotation Matrix (Replies: 0)

  2. Matrix Rotation (Replies: 3)

  3. Rotation matrix (Replies: 5)

  4. Rotation matrix (Replies: 5)

Loading...